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This coupling constant results from expansion of the mixing entropy appearing in the Flory-Huggins (FH) free energy. This latter is a good model (mean-field theory) for describing polymer mixtures of high-molecular weight. However, for those mixtures of low-molecular weight, the FH theory is no longer reliable, and then it would be necessary to investigate the corresponding critical behavior within the framework of the renormalization-group approach applied to the field model (Landau-Ginzburg-Wilson theory) of action (1) with relations (2) and (3). The criticality of the low molecular weight polymer mixture emerges when the temperature parameter t goes to zero and the coupling constant u tends to some universal fixed value u*, which depends only on space dimension d. Usually, this fixed value is calculated using the ε expansion (ε = 4 - d, 4 is the critical dimension of the system).
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note
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Strictly speaking, we can say that the polymer mixture theory constitutes the zeroth order of the so-called loops expansion of the theory of a binary mixture of small molecules or a fluid near the liquid-gas critical point (see Ref. 21).
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